The idea is that the sloshing water can affect the pressure relief valve, I don't believe it can actually cause an issue, but that has always been a concern with domestic hot water heaters (according to the manufacturers)
When I had to move one inside a house to do work behind it, I was always required to drain it first, the theory was that the pressure relief valve could be damaged by water during the movement. I'm not sure how strong any of you people are... but I can't move a 60 gallon water heater that is full anyway, so it was always drained first.
My trailer label states the weight of the fresh water tank when full and the weight of the hot water tank when full, nothing about keeping it empty to travel.